Okay, so I need to compute sin³18° + sin²18°. Let me start by recalling some trigonometric identities that might help here. Hmm, sin²θ and sin³θ... Maybe there's a way to factor this expression? Let's see.

First, let me write down the expression: sin³18° + sin²18°. I can factor out sin²18° from both terms. That would give me sin²18°(sin18° + 1). So, the expression simplifies to sin²18°(sin18° + 1). Hmm, that might be easier to handle. But I need to find the numerical value, so maybe I can compute sin18° first?

Wait, but sin18° is a known value. Let me recall that sin18° is equal to (√5 - 1)/4 multiplied by 2, but I might be mixing up some values here. Let me verify. I remember that sin18° is (sqrt(5)-1)/4, but let me check the exact value. 

Yes, sin18° is indeed (sqrt(5) - 1)/4. Let me confirm that. The exact value of sin18° is (sqrt(5) - 1)/4, which is approximately 0.3090. Let me compute that:

sqrt(5) is approximately 2.236, so sqrt(5) - 1 ≈ 1.236. Divided by 4, that's about 0.309. Yep, that's correct.

So, sin18° ≈ 0.3090. Then sin²18° would be (0.3090)² ≈ 0.0955. And sin³18° would be (0.3090)³ ≈ 0.0295. Adding them together: 0.0295 + 0.0955 ≈ 0.125. Hmm, 0.125 is 1/8. So maybe the exact value is 1/8? Let me check if that's the case.

Alternatively, maybe there's an exact expression without using decimal approximations. Let's try to compute sin³18° + sin²18° exactly. Since sin18° is (sqrt(5)-1)/4, then sin²18° is [(sqrt(5)-1)/4]^2. Let me compute that:

[(sqrt(5) - 1)/4]^2 = (5 - 2sqrt(5) + 1)/16 = (6 - 2sqrt(5))/16 = (3 - sqrt(5))/8.

Similarly, sin³18° is [(sqrt(5)-1)/4]^3. Let's compute that:

[(sqrt(5) - 1)^3]/64. Let me expand the numerator:

(sqrt(5) - 1)^3 = (sqrt(5))^3 - 3*(sqrt(5))^2*1 + 3*sqrt(5)*1^2 - 1^3 = 5*sqrt(5) - 3*5 + 3*sqrt(5) - 1 = 5sqrt(5) - 15 + 3sqrt(5) - 1 = 8sqrt(5) - 16.

So, sin³18° = (8sqrt(5) - 16)/64 = (sqrt(5) - 2)/8.

Therefore, sin³18° + sin²18° = (sqrt(5) - 2)/8 + (3 - sqrt(5))/8. Let's add these two fractions:

[(sqrt(5) - 2) + (3 - sqrt(5))]/8 = (sqrt(5) - 2 + 3 - sqrt(5))/8 = (1)/8. So, the exact value is 1/8, which is 0.125. That matches the decimal approximation I had earlier. So, the answer is 1/8.

Wait, but let me make sure I didn't make any mistakes in expanding (sqrt(5)-1)^3. Let me double-check that step.

Expanding (a - b)^3: a^3 - 3a^2b + 3ab^2 - b^3. So, with a = sqrt(5) and b = 1:

(sqrt(5))^3 = (sqrt(5))^2 * sqrt(5) = 5*sqrt(5)

-3*(sqrt(5))^2*1 = -3*5 = -15

+3*sqrt(5)*1^2 = 3*sqrt(5)

-1^3 = -1

So, adding those up: 5sqrt(5) -15 +3sqrt(5) -1 = (5sqrt(5) + 3sqrt(5)) + (-15 -1) = 8sqrt(5) -16. That's correct. Then divided by 64 gives (8sqrt(5)-16)/64 = (sqrt(5)-2)/8. Correct.

Then sin²18° was (3 - sqrt(5))/8. So adding them:

(sqrt(5)-2)/8 + (3 - sqrt(5))/8 = [sqrt(5) -2 +3 - sqrt(5)]/8 = (1)/8. Perfect! So, the exact value is indeed 1/8. Therefore, the answer is 1/8.

Alternatively, another way to approach this problem is by using trigonometric identities. Let me see if there's another method. For example, since 18° is related to the regular pentagon, perhaps there are some identities or relationships that can be used here. But given that we already found the exact value using the exact expression for sin18°, maybe that's the most straightforward approach.

Alternatively, if I didn't remember the exact value of sin18°, I could use the identity for sin5θ. Let me recall that sin5θ = 16sin^5θ - 20sin³θ + 5sinθ. For θ = 18°, 5θ = 90°, so sin90° = 1. Therefore:

1 = 16sin^518° - 20sin³18° + 5sin18°. Let me set x = sin18°, then the equation becomes:

16x^5 -20x³ +5x -1 =0. Hmm, solving this quintic equation might not be straightforward, but perhaps we can factor it. Let's try x=1: 16 -20 +5 -1=0. So x=1 is a root. Therefore, we can factor out (x-1):

Using polynomial division or synthetic division:

Divide 16x^5 -20x³ +5x -1 by (x -1). Let's set up synthetic division:

Coefficients: 16 (x^5), 0 (x^4), -20 (x^3), 0 (x^2), 5 (x), -1 (constant).

Using root x=1:

Bring down 16.

Multiply by 1: 16. Add to next coefficient: 0 +16=16.

Multiply by1:16. Add to next coefficient: -20 +16= -4.

Multiply by1: -4. Add to next coefficient:0 + (-4)= -4.

Multiply by1: -4. Add to next coefficient:5 + (-4)=1.

Multiply by1:1. Add to constant term: -1 +1=0. So the division works.

Therefore, the polynomial factors as (x -1)(16x^4 +16x³ -4x² -4x +1)=0. So, the other roots are solutions to 16x^4 +16x³ -4x² -4x +1=0. Let me check if this quartic can be factored further. Let's try possible rational roots using Rational Root Theorem. Possible roots are ±1, ±1/2, ±1/4, etc.

Testing x=1: 16 +16 -4 -4 +1=25≠0.

x=-1:16*(-1)^4 +16*(-1)^3 -4*(-1)^2 -4*(-1)+1=16 -16 -4 +4 +1=1≠0.

x=1/2:16*(1/16) +16*(1/8) -4*(1/4) -4*(1/2) +1=1 +2 -1 -2 +1=1≠0.

x=-1/2:16*(1/16) +16*(-1/8) -4*(1/4) -4*(-1/2) +1=1 -2 -1 +2 +1=1≠0.

x=1/4:16*(1/256) +16*(1/64) -4*(1/16) -4*(1/4) +1= (1/16) + (1/4) - (1/4) -1 +1=1/16≈0.0625≠0.

x=-1/4:16*(1/256) +16*(-1/64) -4*(1/16) -4*(-1/4) +1= (1/16) - (1/4) - (1/4) +1 +1= 1/16 - 1/2 +2= approximately 0.0625 -0.5 +2=1.5625≠0.

So no rational roots. Therefore, maybe we can factor the quartic into quadratics. Let's suppose 16x^4 +16x³ -4x² -4x +1 = (ax² +bx +c)(dx² +ex +f). Then:

ad=16

ae + bd=16

af + be + cd= -4

bf + ce= -4

cf=1

Since ad=16, possible pairs for (a,d) are (4,4), (8,2), (16,1), etc. Let's try a=4, d=4.

Then, ae + bd =4e +4b=16 => e + b=4.

af + be + cd=4f + be +4c= -4.

bf + ce= bf + ce= -4.

cf=1.

We need to find integers b,e,c,f such that e + b=4, cf=1. Since cf=1, possible c=1, f=1 or c=-1, f=-1.

Case 1: c=1, f=1.

Then, af + be + cd=4*1 + b*e +4*1=4 + be +4=8 + be= -4 => be= -12.

Also, e + b=4.

So, we have b + e=4 and b*e= -12. Solving quadratic: t^2 -4t -12=0. Discriminant=16 +48=64. Solutions t=(4 ±8)/2=6 or -2. So, b=6, e=-2 or b=-2, e=6.

Check if bf + ce= -2*1 +6*1=4 or 6*1 +(-2)*1=4. But bf + ce= -4. So 4≠-4. Not working.

Case 2: c=-1, f=-1.

Then, af + be + cd=4*(-1) + b*e +4*(-1)= -4 + be -4= be -8= -4 => be=4.

Also, e + b=4.

So, b + e=4 and b*e=4. Quadratic: t^2 -4t +4=0. Solutions t=2, double root. So b=2, e=2.

Then, check bf + ce=2*(-1) +2*(-1)= -2 -2= -4. Which matches the required equation. So this works.

Therefore, the quartic factors as (4x² +2x -1)(4x² +2x -1). Wait, but let's verify:

(4x² +2x -1)(4x² +2x -1)=16x^4 +8x³ -4x² +8x³ +4x² -2x -4x² -2x +1=16x^4 +16x³ -4x² -4x +1. Yes, that's correct. So the quartic factors into (4x² +2x -1)^2=0. Therefore, the roots are solutions to 4x² +2x -1=0. Using quadratic formula:

x = [-2 ± sqrt(4 +16)]/(8) = [-2 ± sqrt(20)]/8 = [-2 ± 2sqrt(5)]/8 = [-1 ± sqrt(5)]/4.

Since sin18° is positive, we take the positive root: (-1 + sqrt(5))/4 ≈ ( -1 + 2.236 )/4 ≈1.236/4≈0.309, which matches our earlier approximation. So, sin18°= (sqrt(5)-1)/4.

Therefore, using this exact value, we can compute sin³18° + sin²18° as we did before, resulting in 1/8. So that's another way to confirm the answer.

Alternatively, maybe there's a way to use angle addition formulas or other identities. Let me think. Since 18° is 30° - 12°, but not sure if that helps. Alternatively, 18° is half of 36°, which is related to the golden ratio. The golden ratio is (1 + sqrt(5))/2, and perhaps that's connected to sin36°, which is 2sin18°cos18°. But maybe that's complicating things.

But since we have already derived the exact value using two different methods (factoring the quintic equation and using the exact expression for sin18°), and both lead to the conclusion that the expression equals 1/8, I can be confident that the answer is indeed 1/8.

Therefore, the final answer is 1/8.
Therefore, the value of $\sin^3 18^\circ + \sin^2 18^\circ$ is $\boxed{\frac{1}{8}}$.
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